Difference between revisions of "Extrapolation"
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<math>O_k = \frac{ \sum_i w_{ki} \, I_i }{\sum_i w_{ki}}; </math> | <math>O_k = \frac{ \sum_i w_{ki} \, I_i }{\sum_i w_{ki}}; </math> | ||
| − | Any method of constructing the weights is valid (as long as <math>\sum_i w_{ki} \neq 0</math>) and depends on the nature of the mapping. A practical use-case consists in performing an interpolation/extrapolation of the data from <math>I</math> to <math>O</math> and thus the weights <math>w</math> are usually a function of the coordinates <math>X</math> and <math>Y</math> and of the norm of the space. More complex techniques could be used, with <math>w</math> also as a function of <math>I</math> and its derivatives, but these will not be considered in this text. | + | Any method of constructing the weights is valid (as long as <math>\sum_i w_{ki} \neq 0 </math> ) and depends on the nature of the mapping. A practical use-case consists in performing an interpolation/extrapolation of the data from <math>I</math> to <math>O</math> and thus the weights <math>w</math> are usually a function of the coordinates <math>X</math> and <math>Y</math> and of the norm of the space. More complex techniques could be used, with <math>w</math> also as a function of <math>I</math> and its derivatives, but these will not be considered in this text. |
==Geometric Weighted Average== | ==Geometric Weighted Average== | ||
Revision as of 10:49, 10 February 2012
Extrapolation is often required when interpolation techniques cannot be used; often when mapping information from one dataset to a grid. Extrapolation methods vary, yielding different results both in quality and in performance. This article proposes a couple of simple extrapolation techniques that use all the information available to extrapolate. The techniques inspire from the weighted average principle.
Contents
Weighted Average
Suppose you have a discrete dataset 
, with coordinates 
on a normed space, 
, and want to map the dataset to another dataset, 
, with a new set of coordinates 
. The operation is performed by constructing mapping coefficients 
, also called "weights", and by performing a linear combination of the elements of with the weights, , to compute each element of .
Any method of constructing the weights is valid (as long as 
) and depends on the nature of the mapping. A practical use-case consists in performing an interpolation/extrapolation of the data from to and thus the weights are usually a function of the coordinates and and of the norm of the space. More complex techniques could be used, with also as a function of and its derivatives, but these will not be considered in this text.
Geometric Weighted Average
The geometric weighted average is the most intuitive approach: it calculates the geometric average of a set of values/points pairs at a given point. The weights are constructed naturally as
